A structure-preserving relaxation Crank-Nicolson finite element method for the Schr\"{o}dinger-Poisson equation

TL;DR

This paper introduces a novel, structure-preserving relaxation Crank-Nicolson finite element method for the Schrödinger-Poisson equation that conserves mass and energy, is computationally efficient, and achieves high accuracy.

Contribution

The paper develops a linear, sequential scheme using a single auxiliary variable to reformulate nonlinear terms, enabling efficient and accurate long-time simulations of the Schrödinger-Poisson system.

Findings

Achieves second-order accuracy in time

Attains $(k+1)$th order spatial accuracy with polynomial degree $k$

Demonstrates superior conservation properties in numerical experiments

Abstract

In this paper, we propose a mass- and modified energy-conservative relaxation Crank-Nicolson finite element method for the Schr\"{o}dinger-Poisson equation. Utilizing only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schr\"{o}dinger equation and the Poisson equation into their equivalent expressions, constructing a system equivalent to the original Schr\"{o}dinger-Poisson equation. Our proposed scheme, derived from this equivalent system, is implemented linearly, avoiding the need for iterative techniques to solve the nonlinear equation. Additionally, it is executed sequentially, eliminating the need to solve a coupled large linear system. We in turn rigorously derive the optimal error estimates for the proposed scheme, demonstrating second order accuracy in time and $(k+1)$th order accuracy in space when employing polynomials of degree up to $k$. Numerical experiments validate the accuracy and effectiveness of our method and emphasize its conservation properties over long-time simulations.